# Questions tagged [normed-spaces]

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### Practical uses of $p$ norms for $p\notin \{1,2,\infty\}$?

We all love normed spaces, but it seems like the $1$-, $2$-, and $\infty$-norms get the lion's share of the love. That's not admittedly not without good reason, but what of the other unsung norms with ...
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### The space of Lipschitz continuous functions is dense in that of uniformly continuous functions?

Let $(X,d)$ be a metric space. Then The space of bounded uniformly continuous functions is dense in that of bounded continuous functions w.r.t. the supremum norm. ref The space of bounded Lipschitz ...
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### For $p\ge 1$, $x\ge 0$ and $y\ge 0$, $|x^{\frac{1}{p}}-y^{\frac{1}{p}}|^p \le |x-y|$?

I vaguely remember seeing an inequality of the form: $$|x^{\frac{1}{p}}-y^{\frac{1}{p}}|^p \le |x-y|$$ for $p\ge 1$, $x\ge 0$ and $y\ge 0$. Is this correct? If so, how is it proven? Can it be ...
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### Derivative of l2-norm w.r.t matrix

I have a matrix A which is size of mm, a vector b which is size of m1. I'd like to derivative of the following function f(A) w.r.t A: $$f(A)=||A*b||_2$$ I need to find the first order Taylor expansion ...
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### Derivative of Frobenius norm with respect to scalar

Can anyone explain to me why $$\frac{d}{d\theta }\left\|\textbf{Aw}-\theta(1-c)\textbf{1} \right\|^{2}= 0$$ is equivalent to $$2(1-c)\textbf{1}^T(\theta(1-c)\textbf{1}-\textbf{Aw})=0$$? I'm not very ...
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### How to prove that normed space is complete?

$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end ...
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### minimal induced norm on sobolev space

I am new to functional analysis and induced norms. I have an exercise question in my university course, and I am trying a lot to get a solution for it. It is a challenging question for me. Could ...
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### Functional analysis - best approximation norm [closed]

I have no idea how to prove or disprove the parts of the below question. I intuitively feel they are correct but cannot form a rigorous mathematical proof for it. Given is a normed space E. Also, ...
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### ways to calculate norm of the Bounded Linear Functional.

$\Lambda$ be a linear functional on $C([0,1])$ defined by $$\Lambda(f) = \int_0^1 xf(x)dx \;\;\; \text{ for } f \in C([0,1]).$$ and use $\|f\|_{sup} = \sup_{x \in [0,1]} |f(x)|$ for $f \in C([0,1])$...
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### On the Interpretation of Volterra's theory of Integral Equations

Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1)$$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous ...
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### Complexity of Lebesgue measurable spaces

Consider a discrete finite set $\Omega=X\times Y \in \mathbb{R}^{m\times n}$ for finite $m,n$. Let $(\Omega,\Sigma,\mu)$ be the measure space. ($\Sigma$ is the power set and $\mu$ is $\sigma$-finite ...
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